Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x-6y &= -1 \\ 3x+6y &= -6\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $6y = -3x-6$ Divide both sides by $6$ to isolate $y$ $y = {-\dfrac{1}{2}x - 1}$ Substitute this expression for $y$ in the first equation. $2x-6({-\dfrac{1}{2}x - 1}) = -1$ $2x + 3x + 6 = -1$ Simplify by combining terms, then solve for $x$ $5x + 6 = -1$ $5x = -7$ $x = -\dfrac{7}{5}$ Substitute $-\dfrac{7}{5}$ for $x$ back into the top equation. $2( -\dfrac{7}{5})-6y = -1$ $-\dfrac{14}{5}-6y = -1$ $-6y = \dfrac{9}{5}$ $y = -\dfrac{3}{10}$ The solution is $\enspace x = -\dfrac{7}{5}, \enspace y = -\dfrac{3}{10}$.